I agree with you re franchise running - Budgens is a better supermarket for it IMO and am also a fan of the branch in Crofton Park.
However, I do not believe that your "chain" analysis of Lordship Lane is incorrect. There is Iceland, Londis, White Stuff, Co-op funeral care, Nicolas, Gourmet Burger Kitchen, Cafe Nero & Favorite (sic) Chicken & Ribs. To name a few off the top of my head.
I think that this shows a mix of chains and independents works well, in fact given I would welcome some (Nicolas, yes please) to Honor Oak with open arms, it could be argued that a mix works better for being so. There is certainly no "thin end of a wedge" argument to be had from one shop.
Whether one agrees with that or not, I would still prefer a parade of chain stores to a row of empty shops, or shudder, shops converted to residences. Those are the eventualities we should avoid at all costs IMO.